1 POWER SERIES 199

spond to two different points instead of a single z\ a stroke of genius if ever
there was one, and which is at the origin of the great theory of Riemann
surfaces, and of their modem generalizations, the complex manifolds which
we shall define in Chapter XVI. The student who wishes to get acquainted
with these beautiful and active theories should read H. Weyl's classic [19]
and the modern presentation by Springer [17] of Riemann surfaces, and
H. Cartan's seminar [7] and the recent book of A. Weil [18] on complex
manifolds.

1. POWER SERIES

In what follows K will denote either the real field R or the complex
field C; its elements will be called scalars. In the vector space K7 over K,
an open (resp. closed) polydisk is a product of p open (resp. closed) balls;
in other words it is a set P defined by conditions of the form \zt — 0j| < rt
(resp. \zl — #;| < rf), !</<;?, on the point z = (zl9 ..., zp), with rL > 0
for every index; a = (ai9..., ap) is the center or P, rls..., rp its radii (a ball
is thus a polydisk having all its radii equal).

(9.1,1) Let P, Q be two open poly disks in Kp such that P n Q 9* 0; for
any two points x, y in P n Q, the segment (Section 8.5) joining x and y is
contained in P n Q; in particular P n Q is connected.

Indeed, if \xi-al\<ri, \yi-ai\<rt, then \txt + (1 - t)yt - a,| <
t\Xt — at\ + (1 — t)\yt — at\ < r, for 0• < / < 1; the last statement follows
from the fact that a segment is connected (by (3.19.1) and (3.197)) and
from (3.19.3),

We introduce the following notation: for any element v «(/?!, ,..,«p)

in Np («| integers >0) and any vector z = (zt,..., zp) e Kp, we write
zv » 2j>zy... Zpp and | v| « /?t + nz + • • * -4- «p. If E is a Banach space

(over K), (rv)vaN^ a family of elements of E having Np as set of indices,
we say that the family (cv zv)v, NJP of elements of E is a power series in p variables

coefficients cv,

(9.1,2) Lei 6 = (5t,..., bf) e Kp be such that bt # 0 for I ^ / < /?, awrf
the family (cvbv) be bounded in E. Then for any system of radii (rf) such that
0<ri< \bi\ for I ^i^p, the power series (cvzv) is normally summable
(7.1) in the closed polydisk of center 0 and radii r, ("Abel's lemma ").uncommon